Problem: Solve for $x$ and $y$ by deriving an expression for $y$ from the second equation, and substituting it back into the first equation. $\begin{align*}2x+6y &= 7 \\ -5x+3y &= 5\end{align*}$
Begin by moving the $x$ -term in the second equation to the right side of the equation. $3y = 5x+5$ Divide both sides by $3$ to isolate $y$ $y = {\dfrac{5}{3}x + \dfrac{5}{3}}$ Substitute this expression for $y$ in the first equation. $2x+6({\dfrac{5}{3}x + \dfrac{5}{3}}) = 7$ $2x + 10x + 10 = 7$ Simplify by combining terms, then solve for $x$ $12x + 10 = 7$ $12x = -3$ $x = -\dfrac{1}{4}$ Substitute $-\dfrac{1}{4}$ for $x$ back into the top equation. $2( -\dfrac{1}{4})+6y = 7$ $-\dfrac{1}{2}+6y = 7$ $6y = \dfrac{15}{2}$ $y = \dfrac{5}{4}$ The solution is $\enspace x = -\dfrac{1}{4}, \enspace y = \dfrac{5}{4}$.